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### Course: Digital SAT Math > Unit 8

Lesson 5: Nonlinear functions: medium# Nonlinear functions | Lesson

A guide to nonlinear functions on the digital SAT

## What are nonlinear functions problems?

### What is a function?

A $f(x)$ , $f$ is the name of the function, $x$ is the input variable, and $f(x)$ is the output.

**function**takes an input and produces an output. In function notation,For example, given $f(x)=2x+1$ , the expression $2x+1$ works as instructions on what to do with the input $x$ . In this case, the input $x$ is multiplied by $2$ , then $1$ is added to the product.

The input of a function can be a , an , or even . Functions can also be .

In this lesson, we'll learn to:

- Evaluate functions algebraically
- Determine inputs and outputs using tables
- Evaluate

**You can learn anything. Let's do this!**

## How do I evaluate functions?

### Evaluate a function given its formula

### Evaluating functions algebraically and using tables

When we encounter an algebraic function, we can find the value of the function at specific inputs. For example, for $f(x)=2x+1$ , we can calculate $f(2)$ , the output of the function $f$ when its input, $x$ , is equal to $2$ .

### What are the steps?

To evaluate a function at a specific input value:

- Plug in the input value for the input variable wherever it appears.
- Perform the operations specified by the function to calculate the ouput.

**Example:**If

With algebraic functions, we can evaluate the function using multiple inputs to create multiple input-output pairs. These input-output pairs can be put in a table, as shown below for $f(x)=2x+1$ .

Sometimes, a table of input-output pairs is provided without an algebraic function. Consider the table below.

The table contains four input-output pairs. We can interpret the information in the table as:

$g(-2)=7$ $g(-1)=0$ $g(0)=4$ $g(1)=-1$

To evaluate a function using a table:

- Find the input value you're looking for in the input column (typically the left column with a header of the input variable such as
).$x$ - Find the corresponding output value in the output column.

**Example:**

Based on the table above, what is the value of $f(3)$ ?

### Try it!

## How do I evaluate composite functions?

### Intro to function composition

### Evaluating composite functions algebraically and using tables

A composite function uses the output of one function as the input of another. For example, for $f(g(x))$ :

is the input to function$x$ .$g$ is the input to function$g(x)$ .$f$

As such, composite functions should be worked from the inside out. Order matters when evaluating composite functions: $g(f(x))$ is not the same as $f(g(x))$ ! For example, for $f(x)={x}^{2}$ and $g(x)=x+1$ , $f(g(1))=4$ , but $g(f(1))=2$ .

To evaluate composite functions at a specific input value:

- Plug in the input value for the input variable wherever it appears in the .
- Perform the operations specified by the
*inner*function to calculate the output. This output becomes the input of the . - Plug in the result of Step 2 for the input variable wherever it appears in the
*outer*function. - Perform the operations specified by the
*outer*function to calculate the final output.

**Example:**If

Composite functions can also be evaluated using a table. The table can have an additional column for a total of three: one column for input and two columns for the outputs of two functions. Consider the table for $f(x)={x}^{2}$ and $g(x)=x+1$ :

From the table, we can tell that $g({1})={2}$ , and $f({2})={4}$ . Therefore, $f(g(1))=4$ .

To evaluate composite functions at a specific input value given a table:

- Find the output value for the
*inner*function corresponding to the specific input value. This is also the input value of the*outer*function. - Find the output value for the
*outer*function corresponding to the input of the result of Step 1.

**Example:**

The table above provides the values of functions $f$ and $g$ at several values of $x$ . What is the value of $g(f(2))$ ?

### Try it!

## How do I compose functions?

### Finding composite functions

### Inputting expressions instead of values into functions

In addition to inputting a specific value, we can also input one function into another function, which creates a composite function defined by a single expression.

For example, for $f(x)={x}^{2}$ and $g(x)=x+1$ , $f(g(x))$ replaces each instance of $x$ in $f$ with $g(x)$ , which is equal to $x+1$ : $f(g(x))=(x+1{)}^{2}$ . Inputting an expression into a function, e.g., $f(x+1),$ works similarly.

A function can also be defined $f(x)={x}^{2}$ and $g(x)=f(x)+1$ , we can replace the $f(x)$ in function $g$ with ${x}^{2}$ : $g(x)={x}^{2}+1$ .

*in terms of*another function. For example, forIf you find yourself struggling to rewrite complex functions, you might want to brush up on the Operations with polynomials and Operations with rational expressions skills, which have their own lessons.

To compose two functions:

- Plug in the expression that defines the
*inner*function wherever the input variable appears in the*outer*function. - Perform the operations specified by the outer function. Combine like terms as needed.

#### Let's look at some examples!

If $f(x)=x-1$ and $g(x)={x}^{2}+1$ , what is $f(g(x))$ ?

If $g(x)={x}^{2}+1$ , what is $g(x-1)$ ?

### Try it!

## Your turn!

## Want to join the conversation?

- MASTERED Alhamdullilah(153 votes)
- yes AlhamduleAllah(9 votes)

- MASTERED Alhamdullilah. Nice videos❤❤❤❤👍👍👍👍👍(46 votes)
- none of that explain the questions involving graphs in the topic test(32 votes)
- fr man, I can't figure out(5 votes)

- There's no lessons on the function graphs(27 votes)
- there is in the next unit(3 votes)

- This is just Higher-Order Function in programming...

After studied college courses and years of experience in Programming, I still have to take SAT as College Entry Exam, what a none-sense...(15 votes)- In India, you have to learn Integrals. As we usually take SAT in the last year, I don't think anybody can do badly on SAT.

I also did college courses from MIT at https://ocw.mit.edu/.(5 votes)

- I'm just curious, guys. Are here mostly international students, or Americans as well?(13 votes)
- I am from Dhaka, Bangladesh! : ) I love Bangladesh.(7 votes)

- it was so easy . At first understoond nothing(14 votes)
- how do i solve function graphs? there isnt any lesson on the function graphs?(9 votes)
- Does anybody know that if we will be provided with digital testing devices in the centre or not in India..??(4 votes)
- no you have to bring your own testing device and if you don't have one then you need to inform college board one month prior to your exam date so that they can arrange it for you(6 votes)

- Hey, while doing the advanced practice for this section, I came across graphs with different sorts of shifts, however, this specific topic doesn't cover them in the videos. Are shifts part of this or will I learn them later on in the future lessons? Would really appreciate a quick response as im really struggling(5 votes)
- you might find the notes regarding shifts of graphs in unit 4 - exponential graphs theory .(3 votes)